2019 School Competition Sprint Round Problems 1−30 School Competition... · Copyright MATHCOUNTS,...

Copyright MATHCOUNTS, Inc. 2018. All rights reserved.

Name

DO NOT BEGIN UNTIL YOU ARE INSTRUCTED TO DO SO.

This section of the competition consists of 30 problems. You will have 40 minutes to complete all the problems. You are not allowed to use calculators, books or other aids during this round. Calculations may be done on scratch paper. All answers must be complete, legible and simplified to lowest terms. Record only final answers in the blanks in the left-hand column of the competition booklet. If you complete the problems before time is called, use the remaining time to check your answers.

In each written round of the competition, the required unit for the answer is included in the answer blank. The plural form of the unit is always used, even if the answer appears to require the singular form of the unit. The unit provided in the answer blank is the only form of the answer that will be accepted.

Total Correct Scorer’s Initials

Founding SponSorS: National Society of Professional Engineers, National Council of Teachers of Mathematics and CNA Foundation

2019School Competition

Sprint RoundProblems 1−30

0

1

2

3

4

5

6

7

8

9

NatioNal SpoNSorS

Raytheon Company U.S. Department of Defense

Northrop Grumman FoundationNational Society of Professional Engineers

CNA InsuranceTexas Instruments Incorporated

3MgivesPhillips 66

Art of Problem SolvingNextThought

2019 MATHCOUNTSNational Competition Sponsor

Copyright MATHCOUNTS, Inc. 2018. All rights reserved. 2019 School Sprint Round

1. _____________

2. _____________

3. _____________

4. _____________

5. _____________

How many integers between 98 and 122 are odd?

In triangle ABC, the measure of A is 40 degrees, and the measure of B is 80 degrees. What is the degree measure of C?

Ms. Rogers surveyed her class by asking, “What is your favorite Skattles candy color?” The graph shows the number of students who selected each Skattles candy color. What is the total number of students who chose red or orange as their favorite Skattles candy color?

If rs

= 22 and the value of r is 242, what is the value of s?

When 12 is subtracted from a number n, the result is one-third of n. What is the value of n?

integers

C

40° 80°

?

BA

degrees

F�� S�� C�� C��

OrangeGreenYellowRedPurple0

2

4

6

1

3

5

C��

N �

��

� S

� ��

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students


6. _____________

7. _____________

8. _____________

9. _____________

10. _____________

The equilateral hexagon shown is formed by joining two equilateral triangles and a square. If this hexagon has perimeter 420 cm, what is its side length?

The figure shows a 3-by-3 section of numbers from a monthly calendar that has five rows and seven columns. What is the value of X + Y + Z + W ?

If q is a rational number such that q divided by 7

8 is 3

5, what is the value of q?

Express your answer as a common fraction.

Audra adds the numbers 2018 and 22, then multiplies the result by 2 and assigns this value to a. Beto multiplies the numbers 2018 and 2, then adds 22 to the result and assigns this value to b. What is the value of a − b?

Alberto has four rectangular tiles. The dimensions of the tiles, in inches, are 2 × 3, 3 × 5, 1 × 4 and 4 × 6. Alberto puts all four tiles together, without gaps or overlaps, to form a single square. What is the side length of the square?

cm

10 12

2624

18

X

W

ZY

//CP: Write all fractions in vertical format; use “divided by” symbol

inches

//CP: If it makes problem easier, could include a picture of the four tiles, but if so, please make it to scale.


11. _____________

12. _____________

13. _____________

14. _____________

15. _____________

If 2x − 1 = 6, what is the value of 4x 2 − 1?

If the operation ⊕ is defined by p ⊕ q = p q2 2+ , what is the value of 8 ⊕ 6?

Flag A and Flag B are composed of a combined total of seven congruent rectangles as shown. What is the ratio of the perimeter of Flag A to the perimeter of Flag B? Express your answer as common fraction.

Jenna’s ice cream shop offers small and large ice cream cones, ten different ice cream flavors and nine different toppings. How many different combinations does Jenna sell with one cone, one ice cream flavor and at most one topping?

Each of the 100 pumpkins decorated by the art club at Squa Middle School was painted, carved or both. If 68 pumpkins were painted and 70 pumpkins were carved, how many pumpkins were painted and carved?

Flag BFlag A

combi-nations

pumpkins


16. _____________

17. _____________

18. _____________

19. _____________

20. _____________

A penny-farthing, like the one shown, is an old-fashioned bicycle from the Victorian era having wheels with different diameters. When the larger wheel makes 20 full revolutions, how many revolutions will the smaller wheel have made if the diameters of the two wheels are 54 inches and 18 inches?

Forrest is playing a four-player card game with his three grandchildren. His three grandchildren are all equally likely to win, but Forrest’s experience and cunning make him twice as likely to win as each of his grandchildren. What is Forrest’s probability of winning? Express your answer as a common fraction.

In regular hexagon ABCDEF, shown here, the degree measure of ABC is x and the degree measure of AEF is y. What is the value of x + y?

Four tokens, numbered as shown, are face-down and shuffled. Kopavy randomly selects two different tokens. If the product of the numbers on his selected tokens is a negative odd number, then he wins a new bicycle. What is the probability that Kopavy will win a new bicycle? Express your answer as a common fraction.

At Freddie’s family reunion, none of her relatives was older than 99 years old and no two relatives were the same age. She found that she had exactly 13 relatives with 2-digit ages where the digits differed by exactly 3. What is the sum of the ages of these 13 relatives?

revo-lutions

//CP: A graphic of a penny-farthing might be a nice touch.

CF

B

D

A

E

x

y

degrees

−8 4 3 −7

years


21. _____________

22. _____________

23. _____________

24. _____________

25. _____________

Bernie played three rounds of golf, and the mean of his scores was 78. If his lowest score was 74. What is the greatest possible value of his highest score?

If (x + y)3 is expanded, what is the sum of the numerical coefficients?

The number line from 1 to 3, shown here, has a scale of 0.1. If a and b represent two consecutive values on this number line, where a < 6 < b, what is the value of a? Express your answer as a decimal to the nearest tenth.

Currently, Lupe and Hector are 61 and 67 years old, respectively. They have two daughters and a grandson, who are 37, 31 and 7 years old, respectively. Lupe noted that their current ages are all prime numbers. During how many of the next 25 years will it also be true that their ages are all prime numbers?

Kermy starts on the lily pad at location A and moves to the lily pad at location B with a series of hops, made only to adjacent lily pads while traveling along the dashed segments shown. If Kermy uses the fewest possible number of hops, how many ways are there for him to get from location A to location B?

1 2 3

years

ways


26. _____________

27. _____________

28. _____________

29. _____________

30. _____________

Mercedes bought some vanilla cupcakes and some chocolate cupcakes. If she

eats one of the vanilla cupcakes, then 1

7 of the remaining cupcakes will be

vanilla. If Mercedes instead eats two of the chocolate cupcakes, then 1

5 of the

remaining cupcakes will be vanilla. How many cupcakes did Mercedes buy?

What is the remainder when 20192019 is divided by 2020?

Ted is taking a true-false test. He has a 90% chance of identifying the correct answer to each question. Unfortunately, there is a 25% chance that he will circle the opposite of his intended answer. What is the probability, expressed as a percent, that Ted circles the correct answer to any particular question on this test?

Frank’s painting is on a large triangular canvas with side lengths 4 meters, 5 meters and 7 meters, as shown. He wants to create a frame such that its inner border is flush with each side of the canvas and every point on its outer border is a distance of exactly 2 cm from the closest point on the canvas. In square centimeters, what is the area of the frame? Express your answer to the nearest whole number.

Chloe charged for admission to her play on three different nights. Each night, a different number of people were in attendance, but remarkably, Chloe collected $541 each night. If the admission charges for each child and each adult were $9 and $17, respectively, how many people in total came to the three showings?

cupcakes

%

5 m

7 m

4 m

cm2

people

2019 School Competition Sprint Round Problems 1−30 School Competition... · Copyright MATHCOUNTS,...

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